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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code> — Mathematical statistics functions</a><ul>
<li><a class="reference internal" href="#averages-and-measures-of-central-location">Averages and measures of central location</a></li>
<li><a class="reference internal" href="#measures-of-spread">Measures of spread</a></li>
<li><a class="reference internal" href="#function-details">Function details</a></li>
<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
<li><a class="reference internal" href="#normaldist-objects"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> objects</a><ul>
<li><a class="reference internal" href="#normaldist-examples-and-recipes"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code> Examples and Recipes</a></li>
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  <div class="section" id="module-statistics">
<span id="statistics-mathematical-statistics-functions"></span><h1><a class="reference internal" href="#module-statistics" title="statistics: Mathematical statistics functions"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statistics</span></code></a> — Mathematical statistics functions<a class="headerlink" href="#module-statistics" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.4.</span></p>
</div>
<p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.8/Lib/statistics.py">Lib/statistics.py</a></p>
<hr class="docutils" />
<p>This module provides functions for calculating mathematical statistics of
numeric (<a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a>-valued) data.</p>
<p>The module is not intended to be a competitor to third-party libraries such
as <a class="reference external" href="https://numpy.org">NumPy</a>, <a class="reference external" href="https://www.scipy.org/">SciPy</a>, or
proprietary full-featured statistics packages aimed at professional
statisticians such as Minitab, SAS and Matlab. It is aimed at the level of
graphing and scientific calculators.</p>
<p>Unless explicitly noted, these functions support <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a>,
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>, <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">Decimal</span></code></a> and <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a>.
Behaviour with other types (whether in the numeric tower or not) is
currently unsupported.  Collections with a mix of types are also undefined
and implementation-dependent.  If your input data consists of mixed types,
you may be able to use <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate"><span class="pre">map()</span></code></a> to ensure a consistent result, for
example: <code class="docutils literal notranslate"><span class="pre">map(float,</span> <span class="pre">input_data)</span></code>.</p>
<div class="section" id="averages-and-measures-of-central-location">
<h2>Averages and measures of central location<a class="headerlink" href="#averages-and-measures-of-central-location" title="Permalink to this headline">¶</a></h2>
<p>These functions calculate an average or typical value from a population
or sample.</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 27%" />
<col style="width: 73%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a></p></td>
<td><p>Arithmetic mean (“average”) of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a></p></td>
<td><p>Fast, floating point arithmetic mean.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.geometric_mean" title="statistics.geometric_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">geometric_mean()</span></code></a></p></td>
<td><p>Geometric mean of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.harmonic_mean" title="statistics.harmonic_mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">harmonic_mean()</span></code></a></p></td>
<td><p>Harmonic mean of data.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a></p></td>
<td><p>Median (middle value) of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a></p></td>
<td><p>Low median of data.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a></p></td>
<td><p>High median of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a></p></td>
<td><p>Median, or 50th percentile, of grouped data.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a></p></td>
<td><p>Single mode (most common value) of discrete or nominal data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.multimode" title="statistics.multimode"><code class="xref py py-func docutils literal notranslate"><span class="pre">multimode()</span></code></a></p></td>
<td><p>List of modes (most common values) of discrete or nomimal data.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.quantiles" title="statistics.quantiles"><code class="xref py py-func docutils literal notranslate"><span class="pre">quantiles()</span></code></a></p></td>
<td><p>Divide data into intervals with equal probability.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="measures-of-spread">
<h2>Measures of spread<a class="headerlink" href="#measures-of-spread" title="Permalink to this headline">¶</a></h2>
<p>These functions calculate a measure of how much the population or sample
tends to deviate from the typical or average values.</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 34%" />
<col style="width: 66%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.pstdev" title="statistics.pstdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">pstdev()</span></code></a></p></td>
<td><p>Population standard deviation of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a></p></td>
<td><p>Population variance of data.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">stdev()</span></code></a></p></td>
<td><p>Sample standard deviation of data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a></p></td>
<td><p>Sample variance of data.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="function-details">
<h2>Function details<a class="headerlink" href="#function-details" title="Permalink to this headline">¶</a></h2>
<p>Note: The functions do not require the data given to them to be sorted.
However, for reading convenience, most of the examples show sorted sequences.</p>
<dl class="function">
<dt id="statistics.mean">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mean" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the sample arithmetic mean of <em>data</em> which can be a sequence or iterable.</p>
<p>The arithmetic mean is the sum of the data divided by the number of data
points.  It is commonly called “the average”, although it is only one of many
different mathematical averages.  It is a measure of the central location of
the data.</p>
<p>If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> will be raised.</p>
<p>Some examples of use:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="go">2.8</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>
<span class="go">2.625</span>

<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">21</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
<span class="go">Fraction(13, 21)</span>

<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.75&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.625&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;0.375&quot;</span><span class="p">)])</span>
<span class="go">Decimal(&#39;0.5625&#39;)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The mean is strongly affected by outliers and is not a robust estimator
for central location: the mean is not necessarily a typical example of
the data points.  For more robust measures of central location, see
<a class="reference internal" href="#statistics.median" title="statistics.median"><code class="xref py py-func docutils literal notranslate"><span class="pre">median()</span></code></a> and <a class="reference internal" href="#statistics.mode" title="statistics.mode"><code class="xref py py-func docutils literal notranslate"><span class="pre">mode()</span></code></a>.</p>
<p>The sample mean gives an unbiased estimate of the true population mean,
so that when taken on average over all the possible samples,
<code class="docutils literal notranslate"><span class="pre">mean(sample)</span></code> converges on the true mean of the entire population.  If
<em>data</em> represents the entire population rather than a sample, then
<code class="docutils literal notranslate"><span class="pre">mean(data)</span></code> is equivalent to calculating the true population mean μ.</p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.fmean">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">fmean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.fmean" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert <em>data</em> to floats and compute the arithmetic mean.</p>
<p>This runs faster than the <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a> function and it always returns a
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.  The <em>data</em> may be a sequence or iterable.  If the input
dataset is empty, raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">fmean</span><span class="p">([</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.25</span><span class="p">])</span>
<span class="go">4.25</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.geometric_mean">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">geometric_mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.geometric_mean" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert <em>data</em> to floats and compute the geometric mean.</p>
<p>The geometric mean indicates the central tendency or typical value of the
<em>data</em> using the product of the values (as opposed to the arithmetic mean
which uses their sum).</p>
<p>Raises a <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if the input dataset is empty,
if it contains a zero, or if it contains a negative value.
The <em>data</em> may be a sequence or iterable.</p>
<p>No special efforts are made to achieve exact results.
(However, this may change in the future.)</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">geometric_mean</span><span class="p">([</span><span class="mi">54</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">36</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">36.0</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.harmonic_mean">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">harmonic_mean</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.harmonic_mean" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the harmonic mean of <em>data</em>, a sequence or iterable of
real-valued numbers.</p>
<p>The harmonic mean, sometimes called the subcontrary mean, is the
reciprocal of the arithmetic <a class="reference internal" href="#statistics.mean" title="statistics.mean"><code class="xref py py-func docutils literal notranslate"><span class="pre">mean()</span></code></a> of the reciprocals of the
data. For example, the harmonic mean of three values <em>a</em>, <em>b</em> and <em>c</em>
will be equivalent to <code class="docutils literal notranslate"><span class="pre">3/(1/a</span> <span class="pre">+</span> <span class="pre">1/b</span> <span class="pre">+</span> <span class="pre">1/c)</span></code>.  If one of the values
is zero, the result will be zero.</p>
<p>The harmonic mean is a type of average, a measure of the central
location of the data.  It is often appropriate when averaging
rates or ratios, for example speeds.</p>
<p>Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60 km/hr.
What is the average speed?</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mi">40</span><span class="p">,</span> <span class="mi">60</span><span class="p">])</span>
<span class="go">48.0</span>
</pre></div>
</div>
<p>Suppose an investor purchases an equal value of shares in each of
three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
What is the average P/E ratio for the investor’s portfolio?</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic_mean</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>  <span class="c1"># For an equal investment portfolio.</span>
<span class="go">3.6</span>
</pre></div>
</div>
<p><a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised if <em>data</em> is empty, or any element
is less than zero.</p>
<p>The current algorithm has an early-out when it encounters a zero
in the input.  This means that the subsequent inputs are not tested
for validity.  (This behavior may change in the future.)</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.6.</span></p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.median">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the median (middle value) of numeric data, using the common “mean of
middle two” method.  If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.
<em>data</em> can be a sequence or iterable.</p>
<p>The median is a robust measure of central location and is less affected by
the presence of outliers.  When the number of data points is odd, the
middle data point is returned:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go">3</span>
</pre></div>
</div>
<p>When the number of data points is even, the median is interpolated by taking
the average of the two middle values:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="go">4.0</span>
</pre></div>
</div>
<p>This is suited for when your data is discrete, and you don’t mind that the
median may not be an actual data point.</p>
<p>If the data is ordinal (supports order operations) but not numeric (doesn’t
support addition), consider using <a class="reference internal" href="#statistics.median_low" title="statistics.median_low"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_low()</span></code></a> or <a class="reference internal" href="#statistics.median_high" title="statistics.median_high"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_high()</span></code></a>
instead.</p>
</dd></dl>

<dl class="function">
<dt id="statistics.median_low">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_low</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_low" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the low median of numeric data.  If <em>data</em> is empty,
<a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.  <em>data</em> can be a sequence or iterable.</p>
<p>The low median is always a member of the data set.  When the number of data
points is odd, the middle value is returned.  When it is even, the smaller of
the two middle values is returned.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go">3</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">median_low</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="go">3</span>
</pre></div>
</div>
<p>Use the low median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.</p>
</dd></dl>

<dl class="function">
<dt id="statistics.median_high">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_high</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_high" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the high median of data.  If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>
is raised.  <em>data</em> can be a sequence or iterable.</p>
<p>The high median is always a member of the data set.  When the number of data
points is odd, the middle value is returned.  When it is even, the larger of
the two middle values is returned.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go">3</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">median_high</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="go">5</span>
</pre></div>
</div>
<p>Use the high median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.</p>
</dd></dl>

<dl class="function">
<dt id="statistics.median_grouped">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">median_grouped</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">interval=1</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.median_grouped" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the median of grouped continuous data, calculated as the 50th
percentile, using interpolation.  If <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>
is raised.  <em>data</em> can be a sequence or iterable.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">52</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">53</span><span class="p">,</span> <span class="mi">54</span><span class="p">])</span>
<span class="go">52.5</span>
</pre></div>
</div>
<p>In the following example, the data are rounded, so that each value represents
the midpoint of data classes, e.g. 1 is the midpoint of the class 0.5–1.5, 2
is the midpoint of 1.5–2.5, 3 is the midpoint of 2.5–3.5, etc.  With the data
given, the middle value falls somewhere in the class 3.5–4.5, and
interpolation is used to estimate it:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go">3.7</span>
</pre></div>
</div>
<p>Optional argument <em>interval</em> represents the class interval, and defaults
to 1.  Changing the class interval naturally will change the interpolation:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">3.25</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">median_grouped</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="n">interval</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="go">3.5</span>
</pre></div>
</div>
<p>This function does not check whether the data points are at least
<em>interval</em> apart.</p>
<div class="impl-detail compound">
<p><strong>CPython implementation detail:</strong> Under some circumstances, <a class="reference internal" href="#statistics.median_grouped" title="statistics.median_grouped"><code class="xref py py-func docutils literal notranslate"><span class="pre">median_grouped()</span></code></a> may coerce data points to
floats.  This behaviour is likely to change in the future.</p>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<ul class="simple">
<li><p>“Statistics for the Behavioral Sciences”, Frederick J Gravetter and
Larry B Wallnau (8th Edition).</p></li>
<li><p>The <a class="reference external" href="https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN">SSMEDIAN</a>
function in the Gnome Gnumeric spreadsheet, including <a class="reference external" href="https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html">this discussion</a>.</p></li>
</ul>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.mode">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">mode</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.mode" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the single most common data point from discrete or nominal <em>data</em>.
The mode (when it exists) is the most typical value and serves as a
measure of central location.</p>
<p>If there are multiple modes with the same frequency, returns the first one
encountered in the <em>data</em>.  If the smallest or largest of those is
desired instead, use <code class="docutils literal notranslate"><span class="pre">min(multimode(data))</span></code> or <code class="docutils literal notranslate"><span class="pre">max(multimode(data))</span></code>.
If the input <em>data</em> is empty, <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> is raised.</p>
<p><code class="docutils literal notranslate"><span class="pre">mode</span></code> assumes discrete data and returns a single value. This is the
standard treatment of the mode as commonly taught in schools:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mode</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="go">3</span>
</pre></div>
</div>
<p>The mode is unique in that it is the only statistic in this package that
also applies to nominal (non-numeric) data:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mode</span><span class="p">([</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">])</span>
<span class="go">&#39;red&#39;</span>
</pre></div>
</div>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 3.8: </span>Now handles multimodal datasets by returning the first mode encountered.
Formerly, it raised <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> when more than one mode was
found.</p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.multimode">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">multimode</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.multimode" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a list of the most frequently occurring values in the order they
were first encountered in the <em>data</em>.  Will return more than one result if
there are multiple modes or an empty list if the <em>data</em> is empty:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">multimode</span><span class="p">(</span><span class="s1">&#39;aabbbbccddddeeffffgg&#39;</span><span class="p">)</span>
<span class="go">[&#39;b&#39;, &#39;d&#39;, &#39;f&#39;]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">multimode</span><span class="p">(</span><span class="s1">&#39;&#39;</span><span class="p">)</span>
<span class="go">[]</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.pstdev">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">pstdev</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pstdev" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the population standard deviation (the square root of the population
variance).  See <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> for arguments and other details.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pstdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
<span class="go">0.986893273527251</span>
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.pvariance">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">pvariance</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">mu=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.pvariance" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the population variance of <em>data</em>, a non-empty sequence or iterable
of real-valued numbers.  Variance, or second moment about the mean, is a
measure of the variability (spread or dispersion) of data.  A large
variance indicates that the data is spread out; a small variance indicates
it is clustered closely around the mean.</p>
<p>If the optional second argument <em>mu</em> is given, it is typically the mean of
the <em>data</em>.  It can also be used to compute the second moment around a
point that is not the mean.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default),
the arithmetic mean is automatically calculated.</p>
<p>Use this function to calculate the variance from the entire population.  To
estimate the variance from a sample, the <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> function is usually
a better choice.</p>
<p>Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if <em>data</em> is empty.</p>
<p>Examples:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="go">1.25</span>
</pre></div>
</div>
<p>If you have already calculated the mean of your data, you can pass it as the
optional second argument <em>mu</em> to avoid recalculation:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">mu</span><span class="p">)</span>
<span class="go">1.25</span>
</pre></div>
</div>
<p>Decimals and Fractions are supported:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;27.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;34.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;41.75&quot;</span><span class="p">)])</span>
<span class="go">Decimal(&#39;24.815&#39;)</span>

<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pvariance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
<span class="go">Fraction(13, 72)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When called with the entire population, this gives the population variance
σ².  When called on a sample instead, this is the biased sample variance
s², also known as variance with N degrees of freedom.</p>
<p>If you somehow know the true population mean μ, you may use this
function to calculate the variance of a sample, giving the known
population mean as the second argument.  Provided the data points are a
random sample of the population, the result will be an unbiased estimate
of the population variance.</p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.stdev">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">stdev</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.stdev" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the sample standard deviation (the square root of the sample
variance).  See <a class="reference internal" href="#statistics.variance" title="statistics.variance"><code class="xref py py-func docutils literal notranslate"><span class="pre">variance()</span></code></a> for arguments and other details.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">stdev</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">3.25</span><span class="p">,</span> <span class="mf">4.75</span><span class="p">])</span>
<span class="go">1.0810874155219827</span>
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.variance">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">variance</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">xbar=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.variance" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the sample variance of <em>data</em>, an iterable of at least two real-valued
numbers.  Variance, or second moment about the mean, is a measure of the
variability (spread or dispersion) of data.  A large variance indicates that
the data is spread out; a small variance indicates it is clustered closely
around the mean.</p>
<p>If the optional second argument <em>xbar</em> is given, it should be the mean of
<em>data</em>.  If it is missing or <code class="docutils literal notranslate"><span class="pre">None</span></code> (the default), the mean is
automatically calculated.</p>
<p>Use this function when your data is a sample from a population. To calculate
the variance from the entire population, see <a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a>.</p>
<p>Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if <em>data</em> has fewer than two values.</p>
<p>Examples:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.75</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="go">1.3720238095238095</span>
</pre></div>
</div>
<p>If you have already calculated the mean of your data, you can pass it as the
optional second argument <em>xbar</em> to avoid recalculation:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="go">1.3720238095238095</span>
</pre></div>
</div>
<p>This function does not attempt to verify that you have passed the actual mean
as <em>xbar</em>.  Using arbitrary values for <em>xbar</em> can lead to invalid or
impossible results.</p>
<p>Decimal and Fraction values are supported:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="k">as</span> <span class="n">D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">([</span><span class="n">D</span><span class="p">(</span><span class="s2">&quot;27.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;30.25&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;34.5&quot;</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="s2">&quot;41.75&quot;</span><span class="p">)])</span>
<span class="go">Decimal(&#39;31.01875&#39;)</span>

<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="k">as</span> <span class="n">F</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">([</span><span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)])</span>
<span class="go">Fraction(67, 108)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This is the sample variance s² with Bessel’s correction, also known as
variance with N-1 degrees of freedom.  Provided that the data points are
representative (e.g. independent and identically distributed), the result
should be an unbiased estimate of the true population variance.</p>
<p>If you somehow know the actual population mean μ you should pass it to the
<a class="reference internal" href="#statistics.pvariance" title="statistics.pvariance"><code class="xref py py-func docutils literal notranslate"><span class="pre">pvariance()</span></code></a> function as the <em>mu</em> parameter to get the variance of a
sample.</p>
</div>
</dd></dl>

<dl class="function">
<dt id="statistics.quantiles">
<code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">quantiles</code><span class="sig-paren">(</span><em class="sig-param">data</em>, <em class="sig-param">*</em>, <em class="sig-param">n=4</em>, <em class="sig-param">method='exclusive'</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.quantiles" title="Permalink to this definition">¶</a></dt>
<dd><p>Divide <em>data</em> into <em>n</em> continuous intervals with equal probability.
Returns a list of <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">-</span> <span class="pre">1</span></code> cut points separating the intervals.</p>
<p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.  Set
<em>n</em> to 100 for percentiles which gives the 99 cuts points that separate
<em>data</em> into 100 equal sized groups.  Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if <em>n</em>
is not least 1.</p>
<p>The <em>data</em> can be any iterable containing sample data.  For meaningful
results, the number of data points in <em>data</em> should be larger than <em>n</em>.
Raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> if there are not at least two data points.</p>
<p>The cut points are linearly interpolated from the
two nearest data points.  For example, if a cut point falls one-third
of the distance between two sample values, <code class="docutils literal notranslate"><span class="pre">100</span></code> and <code class="docutils literal notranslate"><span class="pre">112</span></code>, the
cut-point will evaluate to <code class="docutils literal notranslate"><span class="pre">104</span></code>.</p>
<p>The <em>method</em> for computing quantiles can be varied depending on
whether the <em>data</em> includes or excludes the lowest and
highest possible values from the population.</p>
<p>The default <em>method</em> is “exclusive” and is used for data sampled from
a population that can have more extreme values than found in the
samples.  The portion of the population falling below the <em>i-th</em> of
<em>m</em> sorted data points is computed as <code class="docutils literal notranslate"><span class="pre">i</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">+</span> <span class="pre">1)</span></code>.  Given nine
sample values, the method sorts them and assigns the following
percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.</p>
<p>Setting the <em>method</em> to “inclusive” is used for describing population
data or for samples that are known to include the most extreme values
from the population.  The minimum value in <em>data</em> is treated as the 0th
percentile and the maximum value is treated as the 100th percentile.
The portion of the population falling below the <em>i-th</em> of <em>m</em> sorted
data points is computed as <code class="docutils literal notranslate"><span class="pre">(i</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">(m</span> <span class="pre">-</span> <span class="pre">1)</span></code>.  Given 11 sample
values, the method sorts them and assigns the following percentiles:
0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%.</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="go"># Decile cut points for empirically sampled data</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">105</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">108</span><span class="p">,</span> <span class="mi">92</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span>
<span class="gp">... </span>        <span class="mi">100</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">105</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">119</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="mi">91</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">129</span><span class="p">,</span>
<span class="gp">... </span>        <span class="mi">106</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">74</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">86</span><span class="p">,</span>
<span class="gp">... </span>        <span class="mi">111</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">87</span><span class="p">,</span> <span class="mi">102</span><span class="p">,</span> <span class="mi">121</span><span class="p">,</span> <span class="mi">111</span><span class="p">,</span> <span class="mi">88</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">106</span><span class="p">,</span> <span class="mi">95</span><span class="p">,</span>
<span class="gp">... </span>        <span class="mi">103</span><span class="p">,</span> <span class="mi">107</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">104</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quantiles</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)]</span>
<span class="go">[81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>

</div>
<div class="section" id="exceptions">
<h2>Exceptions<a class="headerlink" href="#exceptions" title="Permalink to this headline">¶</a></h2>
<p>A single exception is defined:</p>
<dl class="exception">
<dt id="statistics.StatisticsError">
<em class="property">exception </em><code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">StatisticsError</code><a class="headerlink" href="#statistics.StatisticsError" title="Permalink to this definition">¶</a></dt>
<dd><p>Subclass of <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> for statistics-related exceptions.</p>
</dd></dl>

</div>
<div class="section" id="normaldist-objects">
<h2><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> objects<a class="headerlink" href="#normaldist-objects" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> is a tool for creating and manipulating normal
distributions of a <a class="reference external" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">random variable</a>.  It is a
class that treats the mean and standard deviation of data
measurements as a single entity.</p>
<p>Normal distributions arise from the <a class="reference external" href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a> and have a wide range
of applications in statistics.</p>
<dl class="class">
<dt id="statistics.NormalDist">
<em class="property">class </em><code class="sig-prename descclassname">statistics.</code><code class="sig-name descname">NormalDist</code><span class="sig-paren">(</span><em class="sig-param">mu=0.0</em>, <em class="sig-param">sigma=1.0</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a new <em>NormalDist</em> object where <em>mu</em> represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic
mean</a> and <em>sigma</em>
represents the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a>.</p>
<p>If <em>sigma</em> is negative, raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a>.</p>
<dl class="attribute">
<dt id="statistics.NormalDist.mean">
<code class="sig-name descname">mean</code><a class="headerlink" href="#statistics.NormalDist.mean" title="Permalink to this definition">¶</a></dt>
<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of a normal
distribution.</p>
</dd></dl>

<dl class="attribute">
<dt id="statistics.NormalDist.median">
<code class="sig-name descname">median</code><a class="headerlink" href="#statistics.NormalDist.median" title="Permalink to this definition">¶</a></dt>
<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Median">median</a> of a normal
distribution.</p>
</dd></dl>

<dl class="attribute">
<dt id="statistics.NormalDist.mode">
<code class="sig-name descname">mode</code><a class="headerlink" href="#statistics.NormalDist.mode" title="Permalink to this definition">¶</a></dt>
<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Mode_(statistics)">mode</a> of a normal
distribution.</p>
</dd></dl>

<dl class="attribute">
<dt id="statistics.NormalDist.stdev">
<code class="sig-name descname">stdev</code><a class="headerlink" href="#statistics.NormalDist.stdev" title="Permalink to this definition">¶</a></dt>
<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Standard_deviation">standard deviation</a> of a normal
distribution.</p>
</dd></dl>

<dl class="attribute">
<dt id="statistics.NormalDist.variance">
<code class="sig-name descname">variance</code><a class="headerlink" href="#statistics.NormalDist.variance" title="Permalink to this definition">¶</a></dt>
<dd><p>A read-only property for the <a class="reference external" href="https://en.wikipedia.org/wiki/Variance">variance</a> of a normal
distribution. Equal to the square of the standard deviation.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.from_samples">
<em class="property">classmethod </em><code class="sig-name descname">from_samples</code><span class="sig-paren">(</span><em class="sig-param">data</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.from_samples" title="Permalink to this definition">¶</a></dt>
<dd><p>Makes a normal distribution instance with <em>mu</em> and <em>sigma</em> parameters
estimated from the <em>data</em> using <a class="reference internal" href="#statistics.fmean" title="statistics.fmean"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmean()</span></code></a> and <a class="reference internal" href="#statistics.stdev" title="statistics.stdev"><code class="xref py py-func docutils literal notranslate"><span class="pre">stdev()</span></code></a>.</p>
<p>The <em>data</em> can be any <a class="reference internal" href="../glossary.html#term-iterable"><span class="xref std std-term">iterable</span></a> and should consist of values
that can be converted to type <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.  If <em>data</em> does not
contain at least two elements, raises <a class="reference internal" href="#statistics.StatisticsError" title="statistics.StatisticsError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">StatisticsError</span></code></a> because it
takes at least one point to estimate a central value and at least two
points to estimate dispersion.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.samples">
<code class="sig-name descname">samples</code><span class="sig-paren">(</span><em class="sig-param">n</em>, <em class="sig-param">*</em>, <em class="sig-param">seed=None</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.samples" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates <em>n</em> random samples for a given mean and standard deviation.
Returns a <a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-class docutils literal notranslate"><span class="pre">list</span></code></a> of <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> values.</p>
<p>If <em>seed</em> is given, creates a new instance of the underlying random
number generator.  This is useful for creating reproducible results,
even in a multi-threading context.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.pdf">
<code class="sig-name descname">pdf</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.pdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Probability_density_function">probability density function (pdf)</a>, compute
the relative likelihood that a random variable <em>X</em> will be near the
given value <em>x</em>.  Mathematically, it is the limit of the ratio <code class="docutils literal notranslate"><span class="pre">P(x</span> <span class="pre">&lt;=</span>
<span class="pre">X</span> <span class="pre">&lt;</span> <span class="pre">x+dx)</span> <span class="pre">/</span> <span class="pre">dx</span></code> as <em>dx</em> approaches zero.</p>
<p>The relative likelihood is computed as the probability of a sample
occurring in a narrow range divided by the width of the range (hence
the word “density”).  Since the likelihood is relative to other points,
its value can be greater than <cite>1.0</cite>.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.cdf">
<code class="sig-name descname">cdf</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.cdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Using a <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative distribution function (cdf)</a>,
compute the probability that a random variable <em>X</em> will be less than or
equal to <em>x</em>.  Mathematically, it is written <code class="docutils literal notranslate"><span class="pre">P(X</span> <span class="pre">&lt;=</span> <span class="pre">x)</span></code>.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.inv_cdf">
<code class="sig-name descname">inv_cdf</code><span class="sig-paren">(</span><em class="sig-param">p</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.inv_cdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the inverse cumulative distribution function, also known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a>
or the <a class="reference external" href="https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/">percent-point</a>
function.  Mathematically, it is written <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">:</span> <span class="pre">P(X</span> <span class="pre">&lt;=</span> <span class="pre">x)</span> <span class="pre">=</span> <span class="pre">p</span></code>.</p>
<p>Finds the value <em>x</em> of the random variable <em>X</em> such that the
probability of the variable being less than or equal to that value
equals the given probability <em>p</em>.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.overlap">
<code class="sig-name descname">overlap</code><span class="sig-paren">(</span><em class="sig-param">other</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.overlap" title="Permalink to this definition">¶</a></dt>
<dd><p>Measures the agreement between two normal probability distributions.
Returns a value between 0.0 and 1.0 giving <a class="reference external" href="https://www.rasch.org/rmt/rmt101r.htm">the overlapping area for
the two probability density functions</a>.</p>
</dd></dl>

<dl class="method">
<dt id="statistics.NormalDist.quantiles">
<code class="sig-name descname">quantiles</code><span class="sig-paren">(</span><em class="sig-param">n=4</em><span class="sig-paren">)</span><a class="headerlink" href="#statistics.NormalDist.quantiles" title="Permalink to this definition">¶</a></dt>
<dd><p>Divide the normal distribution into <em>n</em> continuous intervals with
equal probability.  Returns a list of (n - 1) cut points separating
the intervals.</p>
<p>Set <em>n</em> to 4 for quartiles (the default).  Set <em>n</em> to 10 for deciles.
Set <em>n</em> to 100 for percentiles which gives the 99 cuts points that
separate the normal distribution into 100 equal sized groups.</p>
</dd></dl>

<p>Instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> support addition, subtraction,
multiplication and division by a constant.  These operations
are used for translation and scaling.  For example:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">temperature_february</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>             <span class="c1"># Celsius</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">temperature_february</span> <span class="o">*</span> <span class="p">(</span><span class="mi">9</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">32</span>                     <span class="c1"># Fahrenheit</span>
<span class="go">NormalDist(mu=41.0, sigma=4.5)</span>
</pre></div>
</div>
<p>Dividing a constant by an instance of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> is not supported
because the result wouldn’t be normally distributed.</p>
<p>Since normal distributions arise from additive effects of independent
variables, it is possible to <a class="reference external" href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables">add and subtract two independent normally
distributed random variables</a>
represented as instances of <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a>.  For example:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">birth_weights</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.1</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">,</span> <span class="mf">2.4</span><span class="p">,</span> <span class="mf">2.7</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">drug_effects</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">combined</span> <span class="o">=</span> <span class="n">birth_weights</span> <span class="o">+</span> <span class="n">drug_effects</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">3.1</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">combined</span><span class="o">.</span><span class="n">stdev</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">0.5</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>

<div class="section" id="normaldist-examples-and-recipes">
<h3><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> Examples and Recipes<a class="headerlink" href="#normaldist-examples-and-recipes" title="Permalink to this headline">¶</a></h3>
<p><a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> readily solves classic probability problems.</p>
<p>For example, given <a class="reference external" href="https://nces.ed.gov/programs/digest/d17/tables/dt17_226.40.asp">historical data for SAT exams</a> showing
that scores are normally distributed with a mean of 1060 and a standard
deviation of 195, determine the percentage of students with test scores
between 1100 and 1200, after rounding to the nearest whole number:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">sat</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">1060</span><span class="p">,</span> <span class="mi">195</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span> <span class="o">=</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1200</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">-</span> <span class="n">sat</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1100</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">fraction</span> <span class="o">*</span> <span class="mf">100.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">18.4</span>
</pre></div>
</div>
<p>Find the <a class="reference external" href="https://en.wikipedia.org/wiki/Quartile">quartiles</a> and <a class="reference external" href="https://en.wikipedia.org/wiki/Decile">deciles</a> for the SAT scores:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">round</span><span class="p">,</span> <span class="n">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">()))</span>
<span class="go">[928, 1060, 1192]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">round</span><span class="p">,</span> <span class="n">sat</span><span class="o">.</span><span class="n">quantiles</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)))</span>
<span class="go">[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]</span>
</pre></div>
</div>
<p>To estimate the distribution for a model than isn’t easy to solve
analytically, <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a> can generate input samples for a <a class="reference external" href="https://en.wikipedia.org/wiki/Monte_Carlo_method">Monte
Carlo simulation</a>:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
<span class="gp">... </span>    <span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">11</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>
<span class="gp">...</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">100_000</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">3652260728</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">4582495471</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">)</span><span class="o">.</span><span class="n">samples</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">6582483453</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">quantiles</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">))</span>       
<span class="go">[1.4591308524824727, 1.8035946855390597, 2.175091447274739]</span>
</pre></div>
</div>
<p>Normal distributions can be used to approximate <a class="reference external" href="http://mathworld.wolfram.com/BinomialDistribution.html">Binomial
distributions</a>
when the sample size is large and when the probability of a successful
trial is near 50%.</p>
<p>For example, an open source conference has 750 attendees and two rooms with a
500 person capacity.  There is a talk about Python and another about Ruby.
In previous conferences, 65% of the attendees preferred to listen to Python
talks.  Assuming the population preferences haven’t changed, what is the
probability that the Python room will stay within its capacity limits?</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">750</span>             <span class="c1"># Sample size</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="mf">0.65</span>            <span class="c1"># Preference for Python</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">p</span>         <span class="c1"># Preference for Ruby</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="mi">500</span>             <span class="c1"># Room capacity</span>

<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Approximation using the cumulative normal distribution</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">NormalDist</span><span class="p">(</span><span class="n">mu</span><span class="o">=</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">))</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">0.8402</span>

<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Solution using the cumulative binomial distribution</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">comb</span><span class="p">,</span> <span class="n">fsum</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="n">fsum</span><span class="p">(</span><span class="n">comb</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">r</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">0.8402</span>

<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Approximation using a simulation</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">seed</span><span class="p">,</span> <span class="n">choices</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">seed</span><span class="p">(</span><span class="mi">8675309</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">trial</span><span class="p">():</span>
<span class="gp">... </span>    <span class="k">return</span> <span class="n">choices</span><span class="p">((</span><span class="s1">&#39;Python&#39;</span><span class="p">,</span> <span class="s1">&#39;Ruby&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s1">&#39;Python&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span><span class="p">(</span><span class="n">trial</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">k</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10_000</span><span class="p">))</span>
<span class="go">0.8398</span>
</pre></div>
</div>
<p>Normal distributions commonly arise in machine learning problems.</p>
<p>Wikipedia has a <a class="reference external" href="https://en.wikipedia.org/wiki/Naive_Bayes_classifier#Sex_classification">nice example of a Naive Bayesian Classifier</a>.
The challenge is to predict a person’s gender from measurements of normally
distributed features including height, weight, and foot size.</p>
<p>We’re given a training dataset with measurements for eight people.  The
measurements are assumed to be normally distributed, so we summarize the data
with <a class="reference internal" href="#statistics.NormalDist" title="statistics.NormalDist"><code class="xref py py-class docutils literal notranslate"><span class="pre">NormalDist</span></code></a>:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">height_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">,</span> <span class="mf">5.58</span><span class="p">,</span> <span class="mf">5.92</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">height_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mf">5.5</span><span class="p">,</span> <span class="mf">5.42</span><span class="p">,</span> <span class="mf">5.75</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">weight_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">180</span><span class="p">,</span> <span class="mi">190</span><span class="p">,</span> <span class="mi">170</span><span class="p">,</span> <span class="mi">165</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">weight_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">100</span><span class="p">,</span> <span class="mi">150</span><span class="p">,</span> <span class="mi">130</span><span class="p">,</span> <span class="mi">150</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">foot_size_male</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">12</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">foot_size_female</span> <span class="o">=</span> <span class="n">NormalDist</span><span class="o">.</span><span class="n">from_samples</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">])</span>
</pre></div>
</div>
<p>Next, we encounter a new person whose feature measurements are known but whose
gender is unknown:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="mf">6.0</span>        <span class="c1"># height</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">wt</span> <span class="o">=</span> <span class="mi">130</span>        <span class="c1"># weight</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">fs</span> <span class="o">=</span> <span class="mi">8</span>          <span class="c1"># foot size</span>
</pre></div>
</div>
<p>Starting with a 50% <a class="reference external" href="https://en.wikipedia.org/wiki/Prior_probability">prior probability</a> of being male or female,
we compute the posterior as the prior times the product of likelihoods for the
feature measurements given the gender:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">prior_male</span> <span class="o">=</span> <span class="mf">0.5</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">prior_female</span> <span class="o">=</span> <span class="mf">0.5</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">posterior_male</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_male</span> <span class="o">*</span> <span class="n">height_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>
<span class="gp">... </span>                  <span class="n">weight_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_male</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>

<span class="gp">&gt;&gt;&gt; </span><span class="n">posterior_female</span> <span class="o">=</span> <span class="p">(</span><span class="n">prior_female</span> <span class="o">*</span> <span class="n">height_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">*</span>
<span class="gp">... </span>                    <span class="n">weight_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">wt</span><span class="p">)</span> <span class="o">*</span> <span class="n">foot_size_female</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>
</pre></div>
</div>
<p>The final prediction goes to the largest posterior. This is known as the
<a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation">maximum a posteriori</a> or MAP:</p>
<div class="highlight-pycon3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="s1">&#39;male&#39;</span> <span class="k">if</span> <span class="n">posterior_male</span> <span class="o">&gt;</span> <span class="n">posterior_female</span> <span class="k">else</span> <span class="s1">&#39;female&#39;</span>
<span class="go">&#39;female&#39;</span>
</pre></div>
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